# Louis Bachelier

### Quick Info

Born
11 March 1870
Le Havre, France
Died
26 April 1946
St-Servan-sur-Mer, France

Summary
Louis Bachelier was a French mathematician who is credited with being the first person to model the stochastic process now called Brownian motion.

### Biography

Nul n'est prophete en son pays ... .

The French mathematician, Louis Bachelier is now recognised internationally as the father of financial mathematics, but this fame, which he so justly deserved, was a long time coming. The Bachelier Society, named in his honour, is the world-wide financial mathematics society and mathematical finance is now a scientific discipline of its own. The Society held its first World Congress on 2000 in Paris on the hundredth anniversary of Bachelier's celebrated PhD Thesis Théorie de la Spéculation [24].

Five years before Einstein's famous 1905 paper [4] on Brownian Motion, in which Einstein derived the equation (the partial differential heat/diffusion equation of Fourier) governing Brownian motion and made an estimate for the size of molecules, Bachelier had worked out, for his Thesis, the distribution function for what is now known as the Wiener stochastic process (the stochastic process that underlies Brownian Motion) linking it mathematically with the diffusion equation. The probabilist William Feller had originally called it the Bachelier-Wiener Process. It appears that Einstein in 1905 was ignorant of the work of Bachelier.

Seventy three years before Black and Scholes wrote their famous paper in 1973 [5], Bachelier had derived the price of an option where the share price movement is modelled by a Wiener process and derived the price of what is now called a barrier option (namely the option which depends on whether the share price crosses a barrier). Black and Scholes, following the ideas of Osborne and Samuelson, modelled the share price as a stochastic process known as a Geometric Brownian Motion (with drift).

Louis Bachelier was born in Le Havre in 1870. After education at secondary school in Caen he lost both his parents and had to enter the family business. It was during this period that he seems to have become familiar with the workings of financial markets.

At the age of 22, Bachelier arrived in Paris at the Sorbonne where he followed the lectures of Paul Appell. Joseph Boussinesq and Henri Poincaré (the latter being then aged 38). After some 8 years, in 1900, Bachelier defended his thesis Théorie de la Spéculation before these three men, the favourable report being written by no less a figure than Henri Poincaré, one of the most eminent mathematicians in the world at the time.

Quite what his employment was between 1900 and 1914 (when he was drafted into the French Army during the First World War) is not known. It is known, however, that he received occasional scholarships to continue his studies (on the recommendation of Émile Borel (1871-1956)) and he gave lectures as a 'free professor' at the Sorbonne between 1909 and 1914. One of his courses was Probability calculus with applications to financial operations and analogies with certain questions from physics. In this course he may have drawn out the similarities between the diffusion of probability (the total probability of one being conserved) and the diffusion equation of Fourier (the total heat-energy being conserved). In 1912 he wrote a book Calcul des Probabilités and in 1914 a book Le Jeu, la Chance et le Hazard . At the end of the War he obtained an academic position (lecturer) at Besançon then moved to Dijon (1922), then to Rennes (1925).

In 1926 he tried to go back to Dijon by applying for the vacant chair but was turned down on account of a critical report from Paul Lévy (1886-1971), then a professor aged 40 at the École Polytechnique.

Bachelier in his Thesis, in progressing from a 'drunkards' random walk with $n$ (discrete) steps in time $t$, each step being of length $d$, to a (continuous) distribution for where the drunkard might be at time $t$, realised that there had to be a relationship between $n$ and $d - d$ equal to $(t/n)^{(1/2)}$ for the limit process to 'work'.

In a later paper [38] he showed, effectively, that if a random walk on the $y$-axis is represented as a graph in time with the 'drunkard' making $n$ steps in time $t$, each step of length $d$, the path was such that the tangent of the path angle {i.e. $d$ divided by $(t/n)$} became increasingly large {in the ratio $(n/t)^{(1/2)}$} as $n$ increased. The paths in the time-graph got more and more vertical (up or down) with increasing $n$ but the resulting distribution of where the drunkard might be became increasingly regular. Paul Levy thought that Bachelier had made a mistake in his paper by making the tangent of the path (up or down) constant and Bachelier failed to be appointed at Dijon. Bachelier was furious and wrote to Levy, who, apparently, was unrepentant over this calumny.

The algebraic sum of the upwards and downwards steps taken by the drunkard gives the height of the drunkard at time $t$ above the origin while the sum of the squares of the steps is equal to $t$ and the algebraic and absolute sum of the cubes of the upward and downward steps (and higher powers) become closer and closer to zero. It is these properties of continuity, non-differentiability, infinite 1st order variation, finite 2nd order variation and zero 3rd or higher order variation that gives the drunkard's walk and, in the limit, Brownian Motion some of its unique character and leads to Itô's important Lemma.

It seems extraordinary that Levy was, apparently, unfamiliar with Bachelier's work as Bachelier had by this time (1926) published 3 books and some 13 papers on probability and regarded showing how a continuous distribution could be derived from a discrete distribution as his most important achievement. Levy once told J L Doob that "reading other writers' mathematics gave him physical pain" (see website below) so perhaps it was the case that Levy had never read Bachelier.

Borel, however, must have known Bachelier (he had approved the scholarships to Bachelier). It should be pointed out that Poincaré, who would not have made this mistake over the interpretation of Bachelier's work, had died some 14 years earlier.

It seems that Bachelier, was regarded as being of lesser importance in the eyes of the French mathematical élite (Hadamard, Borel, Lebesgue, Lévy, Baire). His mathematics was not rigorous (it could not be as the mathematical techniques necessary to make it so had not been developed e.g. measure theory and axiomatic probability) although, his results were basically correct.

However, Levy, a few years later, was apparently surprised to find Kolmogorov referring to Bachelier's work. In 1931, Levy wrote a letter of apology to Bachelier and they were reconciled.

Bachelier moved back to Besançon (this time as permanent professor) in 1927 and retired aged 67 in 1937. His last publication was in 1941 and he died in 1946 aged 76.

Bachelier's work is remarkable for herein lie the theory of Brownian Motion (one of the most important mathematical discoveries of the 20th century), the connection between random walks and diffusion, diffusion of probability, curves lacking tangents (non-differentiable functions), the distribution of the Wiener process and of the maximum value attained in a given time by a Wiener process, the reflection principle, the pricing of options including barrier options, the Chapman-Kolmogorov equations in the continuous case,
(namely $f (x_{n} | x_{s}) = \int_{-\infty}^{\infty} f (x_{n} | x_{r}) f (x_{r} | x_{s}) dx_{r}$ where $n > r > s$ where $f$ are the transition densities of a Markov sequence of random variables) and the seeds of Markov Processes, weak convergence of random variables (i.e. convergence in distribution), martingales and Itô stochastic calculus.

Bachelier's treatment and understanding of the theory of Brownian Motion (originally called Brownian Movement) is more elegant and mathematical than in Einstein's 1905 paper. While Einstein had an unsurpassed 'nose' for physics his nose for mathematics was, by his own admission, not so highly developed.

The work of Bachelier leads on to the work of Wiener (1923), Kolmogorov (1931), Itô(1950), and Black, Scholes and Merton (1973).

Bachelier was ahead of his time and his work was not appreciated in his lifetime. In the light of the enormous importance of international derivative exchanges (where the pricing is determined by financial mathematics) the remarkable pioneering work of Bachelier can now be appreciated in its proper context and Bachelier can now be given his proper place.

### References (show)

1. J-M Courtault, Y Kabanov , B Bru , P Crepel , I Lebon and A Le Marchand , Louis Bachelier: On the centenary of Théorie de la Spéculation , Mathematical Finance, 10(3) (2000) 341-353 (see first website listed),
2. M S Taqqu, Bachelier and His Times: A Conversation with Bernard Bru, in Geman H, Madan D, Pliska S R, Vorst T (Eds.) Mathematical Finance - Bachelier Congress 2000, Springer or Finance and Stochastics, 5(1) (2001) 3-32 (contains some 122 references- see second website listed),
3. J-M Courtault, Y Kabanov, (Eds.), Louis Bachelier. Aux origines de la finance mathématique, Presses Universitaires Franc-Comtoises, Besançon (2002)
4. A Einstein, Über die von der molekularkinetishchen Theorie der Warme gefordete Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen, Annalen der Physik , 17 (1905) 549-560. (English Translation:- Einstein A. (1956) Investigations on the Theory of the Brownian Movement, Dover. ['Article On the movement (demanded by the molecular theory of heat) of small particles suspended in a stationary liquid'- see web-site listed].
5. F Black and M Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81 (1973) 637-59.
6. W Feller, An Introduction to Probability Theory and its Applications, Vols. 1 and 2 (1957) Wiley, New York
7. H Geman and M Yor, Pricing and Hedging Double-Barrier Options: A Probabilistic Approach, Math. Finance, 6(4) (1996) 365-78.
8. A N Kolmogorov, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Annalen, 194 (3) (1931) 415-458.
9. R C Merton, Influence of Mathematical Models in Finance on Practice: Past. Present and Future in Howison S D, Kelly F P and Wilmott P (eds,) Mathematical Models in Finance, Chapman Hall. London (1995)
10. M Kac and J L Doob, articles in Bulletin of the American Mathematical Society, 72(1), pt. 2 (1966) 68.
11. M Kac, Probability and Related Topics in Physical Sciences (1959) Interscience, New York.
12. M Lévy, Processus stochastiques et movement brownien (1948) Gauthier-Villars.
13. K Itô, On stochastic differential equations, Memoirs, American Mathematical Society, 4 (1951) 1-51.
14. K Itô and H P McKean, Diffusion Processes and their Sample Paths (1965) Springer.
15. K Itô, Stochastic Differentials, Appl. Math. and Optimization, 1 (1975) 347-81.
16. K Itô, Introduction to Probability Theory (1984) Cambridge University Press (translated from the Japanese),
17. K Itô in Ikeda N, Watanabe S, Fukushima M and Kunita H (eds.), Itô's stochastic calculus and probability theory (1996) Tokyo, ix-xiv.
18. H P McKean, Stochastic Integrals (1960) Academic Press, New York.
19. N Wiener, The mean of a functional of arbitrary elements, Annals of Mathematics, 22 (1920) 66.
20. N Wiener,The average of an analytic functional, Proceedings of the National Academy of Science, 7 (1921) 253.
21. N Wiener, The average of an analytic functional and Brownian motion, Proceedings of the National Academy of Science, 7 (1921) 294.
22. N Wiener, Differentiable Space, Journal of Mathematics and Physics, 2 (1923) 131.
23. N Wiener, The Average Value of a Functional, Proceedings of the London Mathematical Society, 22 (1924) 454.
24. L Bachelier (1900), Théorie de la spéculation, Gauthier-Villars, 70 pp.(see Thesis below).
25. L Bachelier (1912), Calcul des probabilités, Gauthier-Villars, Tome 1, 516 pp.
26. L Bachelier (1914), Le Jeu, la Chance et le Hasard, Bibliothèque de Philosophie scientifique, E.Flammarion, 320 pp.
27. L Bachelier (1937), Les lois des grands nombres du Calcul des Probabilités, Gauthier-Villars,v-vii, 36 pp.
28. L Bachelier (1938), La spéculation et le Calcul des Probabilités, Gauthier-Villars, v-vii, 49 pp.
29. L Bachelier (1939), Les nouvelles méthodes du Calcul des Probabilités, 1939, Gauthier-Villars, v-viii, 69 pp.
30. L Bachelier, Théorie de la Speculation (Thesis), Annales Scientifiques de l'École Normale Superieure (1900) I I I -17, 21-86. (English Translation;- Cootner (ed.), (1964) Random Character of Stock Market Prices, Massachusetts Institute of Technology pp17-78 or Haberman S. and Sibett T. A. (1995) (eds.), History of Actuarial Science, VII, 15-78. London)
31. L Bachelier, Théorie mathématique du jeu, Annales Scientifiques de l'École Normale Supérieure (1901) 143-210.
32. L Bachelier, Théorie des probabilités continues, Journal de Mathématiques Pures et Appliquées (1906) 259-327.
33. L Bachelier, Étude sur les probabilités des causes, Journal de Mathématiques Pures et Appliquées (1908) 395-425.
34. L Bachelier , Le probléme général des probabilités dans les épreuves répétées, Comptes-rendus des Séances de l'Académie des Sciences(1908) 1085-1088.
35. L Bachelier, Les probabilités à plusieurs variables, Annales Scientifiques de l'École Normale Supérieure (1910) 339-360.
36. L Bachelier, Mouvement d'un point ou d'un système matériel soumis à l'action de forces dépendant du hasard, Comptes-rendus des Séances de l'Académie des Sciences (1910) 852-855, présentée par M.H.Poincaré.
37. L Bachelier, Les probabilités cinématiques et dynamiques, Annales Scientifiques de l' École Normale Supérieure (1913) 77-119.
38. L Bachelier, Les probabilités semi-uniformes, Comptes-rendus des Séances de l'Académie des Sciences (1913) 203-205, présentée par M. Appell.
39. L Bachelier, La périodicité du hasard, L'Enseignement Mathématique (1915) 5-11
40. L Bachelier, Sur la théorie des corrélations, Comptes-rendus des Séances de la Société Mathématique de France (1920) 42-44.
41. L Bachelier, Sur les décimales du nombre π, Comptes-rendus des Séances de la Société Mathématique de France (1920) 44-46.
42. L Bachelier, Le probléme général de la statistique discontinue, Comptes-rendus des Séances de l'Académie des Sciences (1923) 1693-1695, présentée par M. d'Ocagne.
43. L Bachelier, Quelques curiosités paradoxales du calcul des probabilités, Revue de Métaphysique et de Morale (1925) 311-320.
44. L Bachelier, Probabilités des oscillations maxima, Comptes-rendus des Séances de l'Académie des Sciences (1941) 836-838.